Foreword to Series xiii Introduction xvii List of Symbols xix Chapter 1. Statistical Properties of a Random Process 1 1.1. Definitions 1 1.1.1. Random variable 1 1.1.2. Random process 2 1.2. Random vibration in real environments 2 1.3. Random vibration in laboratory tests 3 1.4. Methods of random vibration analysis 3 1.5. Distribution of instantaneous values 5 1.5.1. Probability density 5 1.5.2. Distribution function6 1.6. Gaussian random process 7 1.7. Rayleigh distribution 12 1.8. Ensemble averages: through the process 12 1.8.1. n order average 12 1.8.2. Centered moments 14 1.8.3. Variance 14 1.8.4. Standard deviation 15 1.8.5. Autocorrelation function 16 1.8.6. Cross-correlation function 16 1.8.7. Autocovariance 17 1.8.8. Covariance 17 1.8.9. Stationarity 17 1.9. Temporal averages: along the process 23 1.9.1. Mean 23 1.9.2. Quadratic mean rms value 25 1.9.3. Moments of order n 27 1.9.4. Variance standard deviation 28 1.9.5. Skewness 29 1.9.6. Kurtosis 30 1.9.7. Crest Factor 33 1.9.8. Temporal autocorrelation function 33 1.9.9. Properties of the autocorrelation function 39 1.9.10. Correlation duration 41 1.9.11. Cross-correlation 47 1.9.12. Cross-correlation coefficient 50 1.9.13. Ergodicity 50 1.10. Significance of the statistical analysis (ensemble or temporal) 52 1.11. Stationary and pseudo-stationary signals 52 1.12. Summary chart of main definitions 53 1.13. Sliding mean 54 1.14. Test of stationarity 58 1.14.1. The reverse arrangements test (RAT) 58 1.14.2. The runs test 61 1.15 Identification of shocks and/or signal problems 65 1.16. Breakdown of vibratory signal into events : choice of signal samples 68 1.17. Interpretation and taking into account of environment variation 75 Chapter 2. Random Vibration Properties in the Frequency Domain 79 2.1. Fourier transform 79 2.2. Power spectral density 81 2.2.1. Need 81 2.2.2. Definition 82 2.3. Amplitude Spectral Density 89 2.4. Cross-power spectral density 89 2.5. Power spectral density of a random process 90 2.6. Cross-power spectral density of two processes 91 2.7. Relationship between the PSD and correlation function of a process 93 2.8. Quadspectrum cospectrum 93 2.9. Definitions 94 2.9.1. Broadband process 94 2.9.2. White noise 95 2.9.3. Band-limited white noise 95 2.9.4. Narrow band process 96 2.9.5. Colors of noise 97 2.10. Autocorrelation function of white noise 98 2.11. Autocorrelation function of band-limited white noise 99 2.12. Peak factor 101 2.13. Effects of truncation of peaks of acceleration signal on the PSD 101 2.14. Standardized PSD/density of probability analogy 105 2.15. Spectral density as a function of time 106 2.16. Sum of two random processes 106 2.17. Relationship between the PSD of the excitation and the response of a linear system 108 2.18. Relationship between the PSD of the excitation and the cross-power spectral density of the response of a linear system 111 2.19. Coherence function 112 2.20. Transfer function calculation from random vibration measurements 114 2.20.1. Theoretical relations 114 2.20.2. Presence of noise on the input 116 2.20.3. Presence of noise on the response 118 2.20.4. Presence of noise on the input and response 120 2.20.5. Choice of transfer function 121 Chapter 3. Rms Value of Random Vibration 127 3.1. Rms value of a signal as a function of its PSD 127 3.2. Relationships between the PSD of acceleration, velocity and displacement 131 3.3. Graphical representation of the PSD 133 3.4. Practical calculation of acceleration, velocity and displacement rms values 135 3.4.1. General expressions 135 3.4.2. Constant PSD in frequency interval 135 3.4.3. PSD comprising several horizontal straight line segments 137 3.4.4. PSD defined by a linear segment of arbitrary slope 137 3.4.5. PSD comprising several segments of arbitrary slopes 147 3.5. Rms value according to the frequency 147 3.6. Case of periodic signals 149 3.7. Case of a periodic signal superimposed onto random noise 151 Chapter 4. Practical Calculation of the Power Spectral Density 153 4.1. Sampling of signal 153 4.2. PSD calculation methods 158 4.2.1. Use of the autocorrelation function 158 4.2.2. Calculation of the PSD from the rms value of a filtered signal 158 4.2.3. Calculation of PSD starting from a Fourier transform 159 4.3. PSD calculation steps 160 4.3.1. Maximum frequency 160 4.3.2. Extraction of sample of duration T 160 4.3.3. Averaging 167 4.3.4. Addition of zeros 170 4.4. FFT 175 4.5. Particular case of a periodic excitation 177 4.6. Statistical error 178 4.6.1. Origin 178 4.6.2. Definition 180 4.7. Statistical error calculation 180 4.7.1. Distribution of the measured PSD 180 4.7.2. Variance of the measured PSD 183 4.7.3. Statistical error 183 4.7.4. Relationship between number of degrees of freedom, duration and bandwidth of analysis 184 4.7.5. Confidence interval 190 4.7.6. Expression for statistical error in decibels 202 4.7.7. Statistical error calculation from digitized signal 204 4.8. Influence of duration and frequency step on the PSD 212 4.8.1. Influence of duration 212 4.8.2. Influence of the frequency step 213 4.8.3. Influence of duration and of constant statistical error frequency step 214 4.9. Overlapping 216 4.9.1. Utility 216 4.9.2. Influence on the number of degrees of freedom 217 4.9.3. Influence on statistical error 218 4.9.4. Choice of overlapping rate 221 4.10. Information to provide with a PSD 222 4.11. Difference between rms values calculated from a signal according to time and from its PSD 222 4.12. Calculation of a PSD from a Fourier transform 223 4.13. Amplitude based on frequency: relationship with the PSD 227 4.14. Calculation of the PSD for given statistical error 228 4.14.1. Case study: digitization of a signal is to be carried out 228 4.14.2. Case study: only one sample of an already digitized signal is available 230 4.15. Choice of filter bandwidth 231 4.15.1. Rules 231 4.15.2. Bias error 233 4.15.3. Maximum statistical error 238 4.15.4. Optimum bandwidth 240 4.16. Probability that the measured PSD lies between +- one standard deviation 243 4.17. Statistical error: other quantities 245 4.18. Peak hold spectrum 250 4.19. Generation of random signal of given PSD 252 4.19.1. Random phase sinusoid sum method 252 4.19.2. Inverse Fourier transform method 255 4.20. Using a window during the creation of a random signal from a PSD 256 Chapter 5. Statistical Properties of Random Vibration in the Time Domain 259 5.1. Distribution of instantaneous values 259 5.2. Properties of derivative process 260 5.3. Number of threshold crossings per unit time 264 5.4. Average frequency 269 5.5. Threshold level crossing curves 272 5.6. Moments 279 5.7. Average frequency of PSD defined by straight line segments 282 5.7.1. Linear-linear scales 282 5.7.2. Linear-logarithmic scales 284 5.7.3. Logarithmic-linear scales 285 5.7.4. Logarithmic-logarithmic scales 286 5.8. Fourth moment of PSD defined by straight line segments 288 5.8.1. Linear-linear scales 288 5.8.2. Linear-logarithmic scales 289 5.8.3. Logarithmic-linear scales 290 5.8.4. Logarithmic-logarithmic scales 291 5.9. Generalization: moment of order n 292 5.9.1. Linear-linear scales 292 5.9.2. Linear-logarithmic scales 292 5.9.3. Logarithmic-linear scales 292 5.9.4. Logarithmic-logarithmic scales 293 Chapter 6. Probability Distribution of Maxima of Random Vibration 295 6.1. Probability density of maxima 295 6.2. Moments of the maxima probability distribution 303 6.3. Expected number of maxima per unit time 304 6.4. Average time interval between two successive maxima 307 6.5. Average correlation between two successive maxima 308 6.6. Properties of the irregularity factor 309 6.6.1. Variation interval 309 6.6.2. Calculation of irregularity factor for band-limited white noise 313 6.6.3. Calculation of irregularity factor for noise of form G = Const f b 316 6.6.4. Case study: variations of irregularity factor for two narrowband signals 320 6.7. Error related to the use of Rayleigh s law instead of a complete probability density function 321 6.8. Peak distribution function 323 6.8.1. General case. 323 6.8.2. Particular case of narrowband Gaussian process 325 6.9. Mean number of maxima greater than the given threshold (by unit time) 328 6.10. Mean number of maxima above given threshold between two times 331 6.11. Mean time interval between two successive maxima 331 6.12. Mean number of maxima above given level reached by signal excursion above this threshold 332 6.13. Time during which the signal is above a given value 335 6.14. Probability that a maximum is positive or negative 337 6.15. Probability density of the positive maxima 337 6.16. Probability that the positive maxima is lower than a given threshold 338 6.17. Average number of positive maxima per unit of time 338 6.18. Average amplitude jump between two successive extrema 339 6.19. Average number of inflection points per unit of time 341 Chapter 7. Statistics of Extreme Values 343 7.1. Probability density of maxima greater than a given value 343 7.2. Return period 344 7.3. Peak Up expected among p N peaks 344 7.4. Logarithmic rise 345 7.5. Average maximum of p N peaks 346 7.6. Variance of maximum 346 7.7. Mode (most probable maximum value) 346 7.8. Maximum value exceeded with risk 346 7.9. Application to the case of a centered narrowband normal process 346 7.9.1. Distribution function of largest peaks over duration T 346 7.9.2. Probability that one peak at least exceeds a given threshold 349 7.9.3. Probability density of the largest maxima over duration T 350 7.9.4. Average of highest peaks 353 7.9.5. Mean value probability 355 7.9.6. Standard deviation of highest peaks 356 7.9.7. Variation coefficient 357 7.9.8. Most probable value 358 7.9.9. Median 358 7.9.10. Value of density at mode 360 7.9.11. Value of distribution function at mode 361 7.9.12. Expected maximum 361 7.9.13. Maximum exceeded with given risk 361 7.10. Wideband centered normal process 363 7.10.1. Average of largest peaks 363 7.10.2. Variance of the largest peaks 366 7.10.3. Variation coefficient 367 7.11. Asymptotic laws 368 7.11.1. Gumbel asymptote 368 7.11.2. Case study: Rayleigh peak distribution 369 7.11.3. Expressions for large values of p N 370 7.12. Choice of type of analysis 371 7.13. Study of the envelope of a narrowband process 374 7.13.1. Probability density of the maxima of the envelope 374 7.13.2. Distribution of maxima of envelope 379 7.13.3. Average frequency of envelope of narrowband noise 381 Chapter 8. Response of a One-Degree-of-Freedom Linear System to Random Vibration 385 8.1. Average value of the response of a linear system 385 8.2. Response of perfect bandpass filter to random vibration 386 8.3. The PSD of the response of a one-dof linear system 388 8.4. Rms value of response to white noise 389 8.5. Rms value of response of a linear one-degree of freedom system subjected to bands of random noise 395 8.5.1. Case where the excitation is a PSD defined by a straight line segment in logarithmic scales 395 8.5.2. Case where the vibration has a PSD defined by a straight line segment of arbitrary slope in linear scales 401 8.5.3. Case where the vibration has a constant PSD between two frequencies 404 8.5.4. Excitation defined by an absolute displacement 409 8.5.5. Case where the excitation is defined by PSD comprising n straight line segments 411 8.6. Rms value of the absolute acceleration of the response 414 8.7. Transitory response of a dynamic system under stationary random excitation 415 8.8. Transitory response of a dynamic system under amplitude modulated white noise excitation 423 Chapter 9. Characteristics of the Response of a One-Degree-of-Freedom Linear System to Random Vibration 427 9.1. Moments of response of a one-degree-of-freedom linear system: irregularity factor of response 427 9.1.1. Moments 427 9.1.2. Irregularity factor of response to noise of a constant PSD 431 9.1.3. Characteristics of irregularity factor of response 433 9.1.4. Case of a band-limited noise 444 9.2. Autocorrelation function of response displacement 445 9.3. Average numbers of maxima and minima per second 446 9.4. Equivalence between the transfer functions of a bandpass filter and a one-degree-of-freedom linear system 449 9.4.1. Equivalence suggested by D.M. Aspinwall 449 9.4.2. Equivalence suggested by K.W. Smith 451 9.4.3. Rms value of signal filtered by the equivalent bandpass filter 453 Chapter 10. First Passage at a Given Level of Response of a One-Degree-of-Freedom Linear System to a Random Vibration 455 10.1. Assumptions 455 10.2. Definitions 459 10.3. Statistically independent threshold crossings 460 10.4. Statistically independent response maxima 468 10.5. Independent threshold crossings by the envelope of maxima 472 10.6. Independent envelope peaks 476 10.6.1. S.H. Crandall method 476 10.6.2. D.M. Aspinwall method 479 10.7. Markov process assumption 486 10.7.1. W.D. Mark assumption 486 10.7.2. J.N. Yang and M. Shinozuka approximation 493 10.8. E.H. Vanmarcke model 494 10.8.1. Assumption of a two state Markov process 494 10.8.2. Approximation based on the mean clump size 500 Appendix 511 Bibliography 571 Index 591 Summary of Other Volumes in the Series 597

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